The Stacks project

Lemma 33.20.4. Let $k$ be a field. Let $f : X \to Y$ be a morphism of locally algebraic $k$-schemes.

  1. For $y \in Y$, the fibre $X_ y$ is a locally algebraic scheme over $\kappa (y)$ hence all the results of Lemma 33.20.3 apply.

  2. Assume $X$ is irreducible. Set $Z = \overline{f(X)}$ and $d = \dim (X) - \dim (Z)$. Then

    1. $\dim _ x(X_{f(x)}) \geq d$ for all $x \in X$,

    2. the set of $x \in X$ with $\dim _ x(X_{f(x)}) = d$ is dense open,

    3. if $\dim (\mathcal{O}_{Z, f(x)}) \geq 1$, then $\dim _ x(X_{f(x)}) \leq d + \dim (\mathcal{O}_{Z, f(x)}) - 1$,

    4. if $\dim (\mathcal{O}_{Z, f(x)}) = 1$, then $\dim _ x(X_{f(x)}) = d$,

  3. For $x \in X$ with $y = f(x)$ we have $\dim _ x(X_ y) \geq \dim _ x(X) - \dim _ y(Y)$.

Proof. The morphism $f$ is locally of finite type by Morphisms, Lemma 29.15.8. Hence the base change $X_ y \to \mathop{\mathrm{Spec}}(\kappa (y))$ is locally of finite type. This proves (1). In the rest of the proof we will freely use the results of Lemma 33.20.3 for $X$, $Y$, and the fibres of $f$.

Proof of (2). Let $\eta \in X$ be the generic point and set $\xi = f(\eta )$. Then $Z = \overline{\{ \xi \} }$. Hence

\[ d = \dim (X) - \dim (Z) = \text{trdeg}_ k \kappa (\eta ) - \text{trdeg}_ k \kappa (\xi ) = \text{trdeg}_{\kappa (\xi )} \kappa (\eta ) = \dim _\eta (X_\xi ) \]

Thus parts (2)(a) and (2)(b) follow from Morphisms, Lemma 29.28.4. Parts (2)(c) and (2)(d) follow from Lemmas 33.19.3 and 33.19.1.

Proof of (3). Let $x \in X$. Let $X' \subset X$ be a irreducible component of $X$ passing through $x$ of dimension $\dim _ x(X)$. Then (2) implies that $\dim _ x(X_ y) \geq \dim (X') - \dim (Z')$ where $Z' \subset Y$ is the closure of the image of $X'$. This proves (3). $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0B2L. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 33.20.4 as pdf